I'm not a probability theory guy (economics PhD student), but I've recently gotten interested in networks and I've set up a model that I have no idea how to solve. I'm trying to maximize for a scalar and RV which depends on . In order to find first order conditions, I'm trying to solve for satisfying the following:
are iid . and are iid Bernoulli random variables with and . is also a Bernoulli random variable. is a random variable distributed Poisson.
All random variables are independent. is the CDF of .
I'm not really sure how to approach solving this. If you know a method to solve this, a good reference, or that it's simply not possible, I'd be grateful to hear your feedback. I'd like to find an exact solution, if possible.
The first thing I need to ask is if O_i's are discrete random variables and regardless of this answer, what is the domain of the O_i random variables.
If all are discrete, you can get a probability generating function for your Y (even if you don't know the probabilities of O where you represent them symbolically) and then you can impose as many constraints as required to get the actual values of these probabilities.